With the help of WiSE [8], we can obtain the set of all link gains for a specific environment as a function of its geometry. We have observed that in an indoor environment, the link gains deviate from the law of free-space propagation, due to the impacts of reflection, diffraction, and scattering. Furthermore, we have found that obstructed links, that is, those without a clear LOS path, usually undergo more severe attenuation than those with an LOS path, and the added attenuation caused by the obstructions is almost unrelated to the T-R separation distance [9].

Accordingly, the link between a given transmitter and receiver can be classified into one of several different categories according to the number of obstructing objects lying between them. A partition-based path loss analysis for in-home and residential areas at 5.85 GHz was conducted by Durgin et al. in [10]. In this paper, we generalize their framework to distributed wireless networks and propose to estimate both the path loss exponent and attenuation factors using selectively sampled measurements.

Consider the ORBIT testbed, for example, whose layout (top view) is shown in Figure 2. All the links are classified into three categories, namely, links having an LOS path, NLOS links traversing pillars only once, and NLOS links traversing pillars twice (there are no links traversing pillars three or more times in this particular grid). For convenience, in the following we will refer to the links in the three categories as types 0, 1, and 2, respectively.

More generally, let us assume that, in a given network of distributed wireless nodes, there are some paths between node pairs with as many as different types of obstructions. Then, according to our approach, there will be distinct categories (with one LOS category and NLOS categories) for the pathloss formula, where pathloss is the negative dB value of link gain (received power divided by transmitted power). Assume that is a conveniently chosen reference distance, which is typically meter in indoor environments; that is the pathloss at for a single direct ray free-space pathloss; that is the pathloss exponent for the th category; that denotes the pathloss of type at T-R separation distance . A generalized expression for the LOS (type 0) and NLOS (type 1 to type ) pathloss estimate can be given by

where ; is the wavelength; is the pathloss exponent of type links; denotes an added increment resulting from multipath and for obstructions.

Assume  m and that measurements of path loss for links of type are available, that is, . Then the MMSE estimate for the pathloss exponent and attenuations can be obtained by solving

where , ,

denotes the difference between the actual path loss (measured by equipment or emulated using ray tracing tools) and the estimate based on our model in (1).

For those links which are not measured, we can learn their T-R separations as well as their path type (i.e., value of ) through a simple geometric analysis. Then, by plugging the MMSE estimates into (1), the unknown link gains can be predicted.

In order to achieve a good tradeoff between estimation accuracy and the complexity of measurement, an appropriate choice for the sampling set of link gains is important. Unfortunately, this is beyond the scope of classical sampling theory. Therefore, we need to resort to some heuristics.

To begin, let us consider the ORBIT testbed in Figure 2, which shows the 2-D top view of the 400-nodes. The three rectangles in the middle represent the obstructing pillars, and the uniformly-spaced dots denote the possible node locations. Through a simple geometrical calculation, we learned that to have a full diversity of links, that is, types , , and all included, the transmitters have to be placed on one of the 21 locations (within the ellipses highlighted), while the remaining locations do not have type links. Therefore, these 21 transmitter locations have more uncertainty than the remaining ones in terms of link types.

Our design is to measure a total of 1,000 link gains, and to do so by using transmitters at the 21 locations within the highlighted ellipses. Then we randomly choose 350, 600 and 50 samples from links of types , , and , respectively. The numbers of samples are chosen to be proportional to the total number of link gains in each category. This set of choices constitutes a trial. For each trial, we estimate the link gain model parameters via (2), and then substitute them into (1) to obtain the set of link gain estimates, say . For each transmitter-receiver pair on the grid, we employed the ray tracing result from WiSE as our benchmark set of type link gain "measurements", say . The estimation error for a type link is then given by

We repeat the above experiments for 100 trials.

For each type of link, we calculate the bias and standard deviations of , and plot them against the experimental trial index (1, 2,, 100) in (a)-(b) of Figures 3, 4, and 5. (The estimation accuracy of type NLOS links outperforms type links in this example. This is because the number of sampled links versus the number of unmeasured links is greater for type links with respect to our choice, that is, 50/(unmeasured number of type links) 600/(unmeasured number of type links). Basically, the relative estimation accuracy for each category depend on how we allocate the ratio of samples for a given number of measurements. The general principle of selective sampling should guarantee the link gain estimation accuracy for every category.) The common value of in these results was approximately . The solid lines in (a) and (b) denote the empirical bias and standard deviation for the estimation error , respectively; the dashed lines indicate the mean value over the trials. We can conclude from these results that the random choice of link samples can provide sufficient estimation accuracy. (As a rule-of-thumb on testbed experimentation, calibration errors below  dB can be considered quite acceptable.). The method proposed in this paper outperforms spatial interpolation methods. In a separate computation not reported here, we have found that our proposed method, using as many measurements, produces RMS estimation errors 3–4 dB lower than those using spatial interpolation.

Despite the success of the heuristic strategy for the ORBIT testbed, for a more general setup a quantitative or semianalytic approach is desired. The first problem we need to solve is the selection of measurements. Given the size of samples, our objective is to select a most informative subset of link gains. As is traditional, we use entropy as our measure of information since it is a robust measure of the information available from a set of random variables [11]. To this end, let us assume that through site-specific analysis, the relative frequencies of type link gains over the node ensemble of size are known a priori, and are given by , . For links in each category, we characterize their "importance" or entropy by a constant [12]

As a consequence, the entropy of transmitter can be quantized by the weighted sum of the TX-RX links propagating from it, that is,

where

Then the indices of transmitters, , are rearranged according to their entropy, yielding

As a test for the proposed maximum-entropy sampling strategy, we calculated the empirical entropy for all the transmitter locations in Figure 2. It is not surprising that the 21 transmitter locations highlighted in Figure 2 stand out as the ones having the largest entropy.

In light of (8), we can identify the locations for the transmitter-receiver pairs whose link gains are going to be measured. Specifically, assume that is the total number of link gain measurements for a size network, and that we will measure all the link gains between a transmitter, say , and its receivers. We can choose transmitter locations for sampling, which correspond to the first indices in (8). Considering the reciprocity of link gains, is a lower bound for candidate transmitter locations. It is worth noting that spatial correlation is not taken into account by (8). In other words, provided some of the nodes are close enough in space, the adjacent neighbors may exhibit similar entropy values because they are subject to very similar obstruction situations. To remove the redundancy incurred by spatial correlation, we can employ a spatial mask over a sufficiently small area to "filter out" the node with representative entropy value and have it serve as the centroid of a clustered neighborhood.

As shown in Figure 6, the 400-nodes are still arranged into a 20 by 20 array, but the three pillars are reconfigured such that they are of different size and do not align with each other. In this experimental setup, all links are classified into four categories depending on the number of obstructions encountered. Specifically, the LOS links are denoted as type , whereas the NLOS links are grouped into type , type , and type , respectively. As shown in Figure 7, their relative frequencies (normalized by the number of LOS-type inks) are quite different, where the type 1 links significantly outnumber their type 2 and type 3 counterparts, resulting in their discrepancies of entropy.

In this exercise, we will specify measurements, considerably more than the 1,000 measurements for the simpler, more regular ORBIT Lab. This means that there will be transmitting nodes, each sending signals to be measured by the other 399 nodes.

Figures 8(a)–8(c) enumerates the number of type 1, 2, and 3 links for each possible sampling location . Based on Figures 6 and 8, Figure 9 shows the entropy value for each possible transmitter location. It is obvious that our sampling of link gains can focus on a small set of transmitter locations only, namely, those with the largest entropies. The clustering around the spikes (local maxima) can be attributed to the spatial correlation among adjacent transmitters. It can also be seen that there are 20 transmitter indices exhibiting local maximum values of typicality. We select these beacons for our experiment.

By sampling the link gains associated with the 20 transmitter locations (), we can obtain MMSE estimates for the LOS and NLOS pathloss exponents, attenuation factors, and then use them to predict the gains of all the unmeasured links. As a benchmark, we can collect the ensemble of pseudomeasurements given by ray tracing (using WiSE or other tools) and do the same estimation for modeling parameters as above. Figures 10(a), 10(b) and 10(c) compare the difference of these two approaches. The bar on the left corresponds to the full ensemble, which involves all the link gains obtained by ray tracing; while the bar on the right corresponds to the selected samples from the ensemble. Employing the full ensemble of link gains obtained by ray tracing, the estimation for the pathloss exponents and attenuation dB factors are given by ; and , respectively. In contrast, by invoking the proposed "maximum entropy sampling" method, the estimation for the pathloss exponents and attenuation factors are ; and , respectively, which agree well with the previous derivations. We can see from these figures that, although the samples in the selected set are of the ensemble size (), the estimation accuracies for the pathloss parameters using the reduced set are within of the accuracies using the full set. Therefore, the efficacy of our selective sampling approach, which is based on the links' entropy, is verified.

Figure 11 shows a top view of the first floor of the Crawford Hill building [13]. We assume that wireless transceivers (nodes) are deployed in the offices and hallways with uniform randomness. Figure 12 is a scatter plot of pathloss versus distance for the nearly transmit-receive paths among the nodes. The spread is large, but appropriate partitioning of the links can produce individual scatter plots of narrower spread.

Before proceeding, we note that there is a small number of weak links where the pathloss falls below  dB. As a rule-of-thumb, we can ignore any link in this category, since the pathloss is so large that the two nodes can be regarded as "disconnected". As a result, there are link gains to model, of which are LOS (type 0) and are NLOS.

The NLOS links can be further partitioned into links with one or two intervening walls (type 1); links with three or four intervening walls (type 2); and links with more than four intervening walls (type 3). Figure 13 presents fitting parameters ( and ) and RMS fitting errors () for two cases: in (a), all NLOS links are lumped into one category; and in (b), the NLOS links are partitioned, as above, into three categories. It is clear that the refined modeling corresponding to (b) provides a better fit to the scatter plots, since the average is significantly reduced by increasing . Little is gained in this case, however, by increasing beyond 4.

Now assume a target of link gain measurements, that is, about of the total number of link gains. By applying the sampling methodology of Section 3.3, we can pick the five transmitter nodes with maximum entropy and then measure their link gains (with respect to the rest of the network) to estimate the model parameters of (1). The outcome is shown in Figure 14, where the three bar graphs indicate the values of , , and RMS gain estimation error for each of the NLOS categories. The close agreements between the bars for the ensemble and those for the sample sets validate, again, the maximum-entropy approach for selecting transmitters. The RMS gain estimation errors are seen to be dB while, with all NLOS links lumped into one category, this error is close to 10 dB. This demonstrates the significant gain in accuracy by partitioning the NLOS links into several categories.

Most mesh network scenarios will probably have a complexity lying between the two extremes of the ORBIT Lab and the Crawford Hill example, above. In that case, the RMS gain estimation errors for most cases are likely to lie between and  dB. The latter value might be reduced further, not by increasing but by alternative, novel arrangements for choosing the links to be measured. This is a topic for further research.